Hysteretic response of supported drops during forced oscillations

Abstract

Viscous liquid drops undergoing forced oscillations have been shown to exhibit hysteretic deformation under certain conditions both in experiments and by solution of simplified model equations that can only provide a qualitative description of their true response. The first hysteretic deformation results for oscillating pendant drops obtained by solving the full transient, nonlinear Navier–Stokes system are presented herein using a sweep procedure in which either the forcing amplitude A or frequency Ω is first increased and then decreased over a given range. The results show the emergence of turning-point bifurcations in the parameter space of drop deformation versus the swept parameter. For example, when a sweep is carried out by varying Ω while holding A fixed, the first turning point occurs at Ω ≡ Ωu as Ω is being increased and the second one occurs at Ω ≡ Ωl < Ωu as Ω is being decreased. The two turning points shift further from each other and toward lower values of the swept parameter as Reynolds number Re is increased. These turning points mark the ends of a hysteresis range within which the drop may attain either of two stable steady oscillatory states – limit cycles – as identified by two distinct solution branches. In the hysteresis range, one solution branch, referred to as the upper solution branch, is characterized by drops having larger maximum deformations compared to those on the other branch, referred to as the lower solution branch. Over the range Ωl [les ] Ω [les ] Ωu, the sweep procedure enables detection of the upper solution branch which cannot be found if initially static drops are set into oscillation as in previous studies of forced oscillations of supported and captive drops, or liquid bridges. The locations of the turning points and the associated jumps in drop response amplitudes observed at them are studied over the parameter ranges 0.05 [les ] A [les ] 0.125, 20 [les ] Re [les ] 40, and gravitational Bond number 0 [les ] G [les ] 1. Critical forcing amplitudes for onset of hysteresis are also determined for these Re values. The new findings have important ramifications in several practical applications. First, that Ωu − Ωl increases as Re increases overcomes the limitation which is inherent to the current practice of inferring the surface tension and/or viscosity of a bridge/drop liquid from measurement of its resonance frequencies (Chen & Tsamopoulos 1993; Mollot et al. 1993). Moreover, that the value of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has important implications for other free-surface flows such as coating flows.